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Workshop on birational geometry, March 2019
March 25, 2019 15:30–16:30, Moscow, Room 427
 


The Hochschild-Kostant-Rosenberg theorem fails in characteristic pp (after Akhil Mathew)

Vadim Vologodsky

HSE

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Abstract: Let XX be a smooth algebraic variety over a field KK, and let Δ:XX×XΔ:XX×X be the diagonal embedding. Then the cohomology sheaves of the complex LΔΔOXLΔΔOX are canonically identified with the sheaves of differential forms on XX. In particular, there is a spectral sequence from the Hodge cohomology of XX to the hypercohomology of the complex LΔΔOXLΔΔOX . If the characteristic of the base field KK is 00 or larger then dimXdimX, the complex LΔΔOXLΔΔOX is formal, i.e. quasi-isomorphic to the direct sum of its cohomology sheaves. It follows that in this case the above spectral sequence degenerates at the first page. It has been a longstanding question whether this degeneration holds in any characteristic. I will explain a recent result of Akhil Mathew showing that the analogous spectral sequence fails to degenerate for the classifying stack of the finite group scheme μpμp over Fp. This easily yields an example of a smooth projective projective variety X such that the spectral sequence does not degenerate.

Language: English
 
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